To change mapped drive references, you could write a very simply VBScript that would remap these, and execute it as a logon script. Same for the printers.
I would then create a CNAME record on your DNS server pointing oldFileServer to newFileServer (just in case)
EDIT:
You don't change the CNAME. 1) Start up new server. 2) Switch off old server 3) Delete DNS A record for oldFileServer.myDomain.loc 4) Add CNAME record pointing oldFileServer.myDomain.loc to newFileServer.myDomain.loc
As for a sample script, here's one a colleague put together when we migrated our 15 print servers. Just replace the \\oldPrintServer bit, the \\oldPrintServer.myDomain.loc bit, and \\newPrintServer bit. You would then need to call this as a logon script for the user (either via the User Account in AD, from an existing logon script, or via GPO)
Option Explicit
On Error Resume Next
MigratePrint("\\oldPrintServer")
MigratePrint("\\oldPrintServer.myDomain.loc")
Function MigratePrint(strOldServer)
Dim strComputer
Dim strShareName
Dim objWMIService
Dim objPrinter
Dim objItem
Dim colItems
Dim WshNetwork
Dim strNewServer
strComputer = "."
strNewServer="\\newPrintServer"
Set WshNetwork = WScript.CreateObject("WScript.Network")
Set objWMIService = GetObject("winmgmts:\\" & strComputer & "\root\cimv2")
Set colItems = objWMIService.ExecQuery("Select * from Win32_Printer",,48)
For Each objItem in colItems
If objItem.ServerName = strOldServer Then
If objItem.Default = "True" Then
strShareName = objItem.ShareName
End If
WshNetwork.RemovePrinterConnection objItem.ServerName & "\" & objItem.ShareName, True, True
WshNetwork.AddWindowsPrinterConnection strNewServer & "\" & objItem.Sharename
End If
Next
Set objPrinter = CreateObject("WScript.Network")
objPrinter.SetDefaultPrinter (strNewServer & "\" & strShareName)
End Function